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arXiv:1909.12680 · Physical Review A

Conditional probability distributions of finite absorbing quantum walks

Parker Kuklinski

Published Mar 6, 2020• 11 citations
arXivHTMLPDF

Abstract

Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix ${Q}_{n}$, where $n$ is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times $t=O(n)$, fractional quantum revivals for $t=O({n}^{2})$, and stability for $t=O({n}^{3})$. At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.

DOI
10.1103/physreva.101.032309

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Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Qn{Q}_{n}Qn​, where nnn is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times t=O(n)t=O(n)t=O(n), fractional quantum revivals for t=O(n2)t=O({n}^{2})t=O(n2), and stability for t=O(n3)t=O({n}^{3})t=O(n3). At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.
The quantum walk is a unitary analogue to the discrete random walk, and its properties have been increasingly studied since the turn of the millennium. In comparison with the classical random walk, the quantum walk exhibits linear spreading and initial condition dependent asymmetries. As noted early on in the conjecture and subsequent calculation of absorption probabilities in the one dimensional Hadamard walk, the interaction of the quantum walk with an absorbing boundary is fundamentally divergent from classical case. Here, we will survey absorption probabilities for a more general collection of one dimensional quantum walks and extend the method to consider d-dimensional walks in the presence of d-1 dimensional absorbing walls. However, these results are concerned only with local behavior at the boundary in the form of absorption probabilities. The main results of this thesis are concerned with the global behavior of finite quantum walks, which can be described by linear spreading in the short term, modal phenomena in the mid term, and stable distributions in the exceedingly long term. These theorems will be rigorously proved in the one-dimensional case and extrapolated to higher dimensional quantum walks. To this end we introduce QWSim, a new and robust computational engine for displaying finite two dimensional quantum walks.
Absorbing boundaries introduce non-unitary dynamics in quantum mechanical systems. In particular, it has been shown that absorbing boundaries partially reflect quantum information in a variety of one-dimensional quantum walks. The methods used to compute absorption probabilities for quantum walks can be extended to calculate absorption probabilities for general finite and discrete quantum mechanical systems. These methods can also be used to compute moments of the associated absorption time distributions. We conclude the paper by discussing a modification of the quantum Zeno effect in these absorption systems.
Abstract Absorption of two-state coined quantum walks on a finite line with two sinks located at N and − N is investigated. Elaborating on the results of Konno et al (2003 J. Phys. A: Math. Gen. 36 241), we derive closed formulas for the absorption probabilities at the boundaries in the limit of large system size N . Two limiting cases are considered, with the starting position k being independent of N , or kept at a constant distance δ from one of absorbers. In the first scenario, the absorption probability is determined only by the coin parameter and polar angle of the initial coin state decomposed into the eigenbasis of the coin operator. In the second case, a correction depending exponentially on δ is introduced. Finally, we perform an extensive numerical investigation for small system size N , showing excellent agreement between numerical and analytical results.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.
We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.
The dynamics of the survival probability of quantum walkers on a one-dimensional lattice with random distribution of absorbing immobile traps is investigated. The survival probability of quantum walkers is compared with that of classical walkers. It is shown that the time dependence of the survival probability of quantum walkers has a piecewise stretched exponential character depending on the density of traps in numerical and analytical observations. The crossover between the quantum analogues of the Rosenstock and Donsker–Varadhan behavior is identified.